THE VOLUME CONJECTURE OF KNOTS AND ITS RAMIFICATIONS

Kashaev，J.Murakami和H.Murakami提出的纽结的体积猜想指出，有色Jones多项式的渐近性质是著名的Jones多项式的推广，它决定了纽结的补的单纯体积。通过H.Murakami，J.Murakami，M.Okamoto，T.T.Takata和作者的计算机实验，这个猜想现在被推广到涉及Chern-Simons不变量，许多数学家现在对这个问题感兴趣.研究的目的是研究纽结补的几何和有色Jones多项式之间的关系，我们已经证明了有色Jones多项式控制Neuman-Zagler-Yoshida函数和A-多项式.实际上，通过对体积猜想的研究，我们从有色Jones多项式得到一个势函数，它可以变形为Neuman-Zagler-Yoshida函数，它的偏微分方程组给出了A-多项式，Neuman-Zagler-Yoshida函数定义在尖点双曲流形的双曲结构的变形空间上，描述了体积与Chern-Simons不变量之间的解析关系。纽结的A-多项式是从纽结的基本群的表示空间中定义的，在Dehn外科理论中起着重要的作用。最近，一些数学家也对A-多项式感兴趣，因为它的马勒测度给出了一定的DedekindZeta函数。作者被邀请参加了许多会议，并有很多机会展示我们的结果。作者希望许多数学家对这一研究感兴趣。

The volume conjecture of knots states that the asymptotic behavior of the colored Jones polynomials, a generalization of the famous Jones polynomial, determines the simplicial volume of the complement of a knot, which is proposed by Kashaev, J. Murakami and H. Murakami. This conjecture is now generalized to involve the Chern-Simons invariant through the computer experiment by H. Murakami, J. Murakami, M. Okamoto, T. Takata and the author, and many mathematicians are now interested in this problem.The purpose of this research is to investigate the relationship between the geometry of the knot complement and the colored Jones polynomials, and we have shown that the colored Jones polynomials dominate both the Neuman-Zagler-Yoshida function and the A-polynomial. In fact, from the study of the volume conjecture, we derive a potential function from the colored Jones polynomials which can be deformed into the Neuman-Zagler-Yoshida function and whose partial differential equations gives the A-polynomial.The Neuman-Zagler-Yoshida function is defined over deformation space of the hyperbolic structures of cusped hyperbolic manifolds and describes the analytic relation between the volumes and the Chern-Simons invariants. The A-polynomial of knots is defined from the representation space of the fundamental groups of knots and plays an important role in the theory of Dehn surgery. Recently, some number theorists are also interested in the A-polynomial because its Mahler measure gives certain Dedekind zeta function.The author was invited to many conferences and had many opportunities to exhibit our results. The author hopes many mathematicians are interested in this research.